University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

339

Pages:

2
Author:

Luciano Kirby

Question 1 Advanced Math Which one is a simplified expression for A(A' + B')(B+C)(B+C' + D) Answer Step 1 Multiply A with the terms in the first bracket: = A(A'+Β')(B+C' + D) = [V' + AB'] (B+C)(B+C' + D) Since PP' 0 = [0+ AB'] (B+C)(B+C' + D) = AB'(B+C)(B+C' + D) Step 2 Multiply AB' with the terms in the first bracket of the obtained expression: = [AB'(B+C)] (B+C' + D) = [ABB' + AB'C] (B+C' + D) = [A(BB') + AB'C] (B+C' + D) Since PP' = 0 = [0+ AB'C] (B+C' + D) = AB'C(B + C' + D) Step 3 Multiply AB'C with all the terms in the bracket of the obtained expression: = AB'C(B+C' + D) = ABB'C + АВ'СС' + AB'CD = А(BB')C + AB'(СС') + AB'C'D Since PP' = 0 = 0 + 0 + AB'CD = AB'CD Thus the simplified expression is AB'CD. Question 2 In fall 2014, 38% of applicants with a Math SAT of 700 or more were admitted by a certain university, while 12% with a Math SAT of less than 700 were admitted. Further, 32% of all applicants had a Math SAT score of 700 or more. What percentage of admitted applicants had a Math SAT of 700 or more? (Round your answer to the nearest percentage point.) Answer Step 1 Let A - Percent math S A T score of less than 700 or more in all applicants. B - Percent math SAT score of 700 or more than admitted applicants. C - Percent math SAT score of less than 700 in admitted applicants. Step 2 Formula used P(B|A) = P(A|B)P(B) / (P(A|C)P(C) + (P(A|B)P(B)) ...(i) Step 3 Given P(A|B) = 32% = 0.32 P(B) = 38% = 0.38 P(A|C) = 1 - 0.32 = 0.68 P(C) = 12% = 0.12 Step 4 Put these values in(i) P(B|A) = (0.32)(0.38) / (0.68)(0.12)+((0.32)(0.38)) = 0.1216 / 0.0816+0.1216 = 0.1216 / 0.2032 ≈ 0.598425 Step 5 Percentage of admitted applicants had a Math SAT of 700 or more is 59.84% ≈ 60% Question 3 Express the following propositions in symbolic form, and identify the principal connective: i.Either Karen is studying computing and Ketsi is not studying mathematics, or Ketsi is studying mathematics. ii.lt is not the case case that if it is sunny then i will carry an umbrella. Answer Step 1 (i) Let, p:Karen is studying computing. q:Ketsi is studying mathematics. q:Ketsi is now studying mathematics. Then,(p^!q) V q:Either Karen is studying computing and Ketsi is not mathematics, or Ketsi is studying mathematics. Here, principal connective is "vv" (i.e."or"). Step 2 (ii) Let, p: It is sunny. q: I will carry an umberella. Then, !(p→q): It is not the case that if it is sunny then i will carry an umbrella. Here, principal connective is ""(i.e."not").